This study describes a split population logit model that can be useful to researchers who are modeling a binary dependent variable that is measured with a biased instrument. To motivate the study we identify two common, yet widely unrecognized, circumstances in which political scientists are likely to study dichotomous variables that have been measured with bias. In one such setting (e.g., surveys) the strategic interests of actors will lead them to misrepresent an attitude or behavior. In another such setting (e.g., content analysis of events) researchers' instruments are unable to distinguish between the absence of a characteristic or event and missing data. We briefly argue that "unobservability," "zero-inflated," and other models form a single class of models that allow researchers to model the bias in operational instruments, and thus not only correct bias in statistical inference but, more importantly, produce theoretical accounts of the bias and then test the hypotheses that those accounts imply. We derive the likelihood function for the split population logit model, describe the properties of its MLEs, present the results from a Monte Carlo study, and briefly describe code that researchers can use to implement the model in the Stata statistical package.